Generalized quantum data-syndrome codes and belief propagation decoding for phenomenological noise

TL;DR

This paper introduces generalized quantum data-syndrome codes with belief propagation decoding to improve error correction in quantum memory, effectively handling phenomenological noise with fewer syndrome extraction rounds.

Contribution

It develops a unified check matrix for diverse error sources and designs BP algorithms applicable to general sparse quantum codes, enhancing error correction performance.

Findings

Achieved over 3% error threshold with BP decoding on rotated toric codes.

Fewer syndrome extraction rounds suffice for higher code distances.

BP decoding remains effective even with high syndrome error rates.

Abstract

Quantum stabilizer codes often struggle with syndrome errors due to measurement imperfections. Typically, multiple rounds of syndrome extraction are employed to ensure reliable error information. In this paper, we consider phenomenological decoding problems, where data qubit errors may occur between extractions, and each measurement can be faulty. We introduce generalized quantum data-syndrome codes along with a generalized check matrix that integrates both quaternary and binary alphabets to represent diverse error sources. This results in a Tanner graph with mixed variable nodes, enabling the design of belief propagation (BP) decoding algorithms that effectively handle phenomenological errors. Importantly, our BP decoders are applicable to general sparse quantum codes. Through simulations, we achieve an error threshold of more than 3\% for quantum memory protected by rotated toric codes, using solely BP without post-processing. Our results indicate that $d$ rounds of syndrome extraction are sufficient for a toric code of distance $d$. We observe that at high error rates, fewer rounds of syndrome extraction tend to perform better, while more rounds improve performance at lower error rates. Additionally, we propose a method to construct effective redundant stabilizer checks for single-shot error correction. Our simulations show that BP decoding remains highly effective even with a high syndrome error rate.